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Spectral Radius of Graphs provides a thorough overview of important results on the spectral radius of adjacency matrix of graphs that have appeared in the literature in the preceding ten years, most of them with proofs, and including some previously unpublished results of the author. The primer begins with a brief classical review, in order to provide the reader with a foundation for the subsequent chapters. Topics covered include spectral decomposition, the Perron-Frobenius theorem, the Rayleigh quotient, the Weyl inequalities, and the Interlacing theorem. From this introduction, the book delves deeper into the properties of the principal eigenvector; a critical subject as many of the results on the spectral radius of graphs rely on the properties of the principal eigenvector for their proofs. A following chapter surveys spectral radius of special graphs, covering multipartite graphs, non-regular graphs, planar graphs, threshold graphs, and others. Finally, the work explores results on the structure of graphs having extreme spectral radius in classes of graphs defined by fixing the value of a particular, integer-valued graph invariant, such as: the diameter, the radius, the domination number, the matching number, the clique number, the independence number, the chromatic number or the sequence of vertex degrees.
Throughout, the text includes the valuable addition of proofs to accompany the majority of presented results. This enables the reader to learn tricks of the trade and easily see if some of the techniques apply to a current research problem, without having to spend time on searching for the original articles. The book also contains a handful of open problems on the topic that might provide initiative for the reader's research.
- Dedicated coverage to one of the most prominent graph eigenvalues
- Proofs and open problems included for further study
- Overview of classical topics such as spectral decomposition, the Perron-Frobenius theorem, the Rayleigh quotient, the Weyl inequalities, and the Interlacing theorem
Math researchers and advanced students in graph theory, linear algebra, related fields
Chapter 1: Introduction
- 1.1 Graphs and Their Invariants
- 1.2 Adjacency Matrix, its Eigenvalues, and its Characteristic Polynomial
- 1.3 Some Useful Tools from Matrix Theory
Chapter 2: Properties of the Principal Eigenvector
- 2.1 Proportionality Lemma and the Rooted Product
- 2.2 Principal Eigenvector Components Along a Path
- 2.3 Extremal Components of the Principal Eigenvector
- 2.4 Optimally Decreasing Spectral Radius By Deleting Vertices or Edges
- 2.5 Regular, Harmonic, and Semiharmonic Graphs
Chapter 3: Spectral Radius of Particular Types of Graph
- 3.1 Nonregular Graphs
- 3.2 Graphs with a Given Degree Sequence
- 3.3 Graphs with A Few Edges
- 3.4 Complete Multipartite Graphs
Chapter 4: Spectral Radius and Other Graph Invariants
- 4.1 Selected Autographix Conjectures
- 4.2 Clique Number
- 4.3 Chromatic Number
- 4.4 Independence Number
- 4.5 Matching Number
- 4.6 The Diameter
- 4.7 The Radius
- 4.8 The Domination Number
- 4.9 Nordhaus-Gaddum Inequality for the Spectral Radius
- No. of pages:
- © Academic Press 2014
- 23rd September 2014
- Academic Press
- Paperback ISBN:
- eBook ISBN:
Mathematical Institute, Serbian Academy of Science and Arts, Belgrade, Serbia and University of Primorska, Koper, Slovenia.
"It covers topics of great interest which are attractive not only to researchers in graph theory, but also to other specialists. Therefore, especially for a researcher in the field, this monograph is a must-buy!" --Zentralblatt MATH
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